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Theory of General Invariance (Postulate I: \( [\hat{x}_{i}, \hat{x}_{j}] =…
Theory of General Invariance
Postulate I: \( [\hat{x}_{i}, \hat{x}_{j}] = il_{p}^{2}\hat{E}\) where \( \hat{E} \) is a 0-1 adjacency matrix (which we call the entanglement field operator).
In the \( Cl(\mathbb{R}, 3) \) basis \( e_{1}e_{2}e_{3}l_{p}^{2}\hat{E} \) is a matrix-valued trivector (volume element).
In the \( Cl(\mathbb{R}, 2) \) basis \( e_{1}e_{2}l_{p}^{2}\hat{E} \) is a matrix-valued bivector (area element).
Generalized Uncertainty Principle: \( \sigma_{i}\sigma_{j} \geq \dfrac{l_{p}^{2}}{2}|<\hat{E}>|^{2} \) where \( 0 \leq \ <\hat{E}> \ \leq 1 \).
emergent arrow of time: \( <\hat{E}> \longrightarrow 0 \) (sparse limit).
Smooth manifold in the sparse limit.
Postulate 3: The operator-valued power spectrum \( \mathcal{P}[\hat{A}(\mathcal{G})] \equiv |G> \) generates a locally finite-dimensional Hilbert space with the metric eigenfunction \( |g> = |g_{+}> + |g_{-}> \).
The Entanglement Field Equation: \( |G> = |g_{+}> + |g_{-}> \)
Postulate 2: For every linear self-adjoint operator \( \hat{A} \) there exists an underlying graph \( \mathcal{G} \).
All QM operators become adjacency operators \( \hat{A}(\mathcal{G}) \) with \( \mathcal{G} \) specifying the microscopic structure of spacetime.
\( \mathcal{G} \) has two unique states: superimposed (degrees of freedom optimally compressed) and collapsed (optimally decompressed).
Postulate 2a: When spacetime is in superposition, the vertices \( (e_{1}, e_{2} ) \) of \( \mathcal{G} \) are elements of the Geometric (Clifford) algebra \( Cl(\mathbb{R}, 2) \); the imaginary constant is a bivector (area element) \( i = e_{1}e_{2} \).
The geometrized phase space is \( (\hat{p} \cdot \sigma, \hat{q} \cdot \sigma )\) (which we call
qubitspace
).
SU(2) X SU(2)
The momentum \( \hat{p} = -e_{1}e_{2}\hbar \nabla \) is an operator-valued bivector (area element).
The position \( \hat{q} = \hat{x}e_{1} + \hat{y}e_{2} + \hat{z}e_{3} \) is an operator-valued vector (invariant w.r.t basis change).
Postulate 2b: When spacetime is
not
in superposition, the vertices \( (e_{1}, e_{2}, e_{3} )\) of \( \mathcal{G} \) are elements of the Geometric (Clifford) algebra \( Cl(\mathbb{R}, 3) \); the imaginary constant is a trivector (volume element) \( i = e_{1}e_{2}e_{3} \).
The geometrized phase space is \( ( \hat{p} \otimes \hat{p}, \hat{q} \otimes \hat{q}) \) (which we call
bitspace
).
SO(3) X SO(3)
The momentum \( \hat{p} = -e_{1}e_{2}e_{3}\hbar \nabla \) is an operator-valued trivector (volume element).
The position \( \hat{q} = \hat{x}e_{1} + \hat{y}e_{2} + \hat{z}e_{3} \) is an operator-valued vector (invariant w.r.t basis change).
Holographic Flow: Transitions between the \( Cl(\mathbb{R}, 3) \) and \( Cl(\mathbb{R}, 2) \) basis generates the \( SO(4) \longrightarrow SO(3) \times SO(3) \) \( SU(2) \times SU(2) \longrightarrow SO(4) \) representation of 4-dim Euclidean space \( \mathbb{R}^{4} \).
Emergent Time
Principle of Least Description
: Nature will choose
either
a smooth manifold
or
a Hilbert space as an underlying mathematical structure.
Principle of Scale Invariance
: the laws of physics do not depend on scale.
Principle of Finiteness
: the laws of physics strictly forbid infinite quantities.
Principle of General Invariance
: the laws of physics are scale-invariant within a fundamental UV/IR regime.
Principle of Least Description + Principle of Finiteness + Principle of Scale Invariance = Principle of General Invariance